Solving $x^2 - 12x + 61 = 0$ Using The Quadratic Formula

Introduction

The quadratic formula is a powerful tool in algebra for finding the solutions (or roots) of any quadratic equation. A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a eq 0$. The quadratic formula provides a straightforward method to solve for $x$, even when the equation cannot be easily factored. In this article, we will delve into the application of the quadratic formula to solve the specific equation $x^2 - 12x + 61 = 0$. This comprehensive guide will walk you through each step, ensuring a clear understanding of the process and the underlying concepts. Understanding the quadratic formula is crucial not only for academic success but also for various real-world applications, including physics, engineering, and economics. The formula is a cornerstone of algebraic problem-solving and a testament to the elegance of mathematical solutions. Mastering it can significantly enhance your mathematical toolkit and confidence. We will start by understanding the significance of the quadratic formula, then proceed to identify the coefficients in our given equation, and finally, apply the formula step-by-step to find the solutions. This approach ensures that by the end of this guide, you will be well-versed in solving quadratic equations using this method.

Understanding the Quadratic Formula

The quadratic formula is derived from the general form of a quadratic equation, $ax^2 + bx + c = 0$, using a method called completing the square. This formula is expressed as:

$x = \frac{-b pm psqrt{b^2 - 4ac}}{2a}$

In this formula, $a$, $b$, and $c$ are the coefficients of the quadratic equation. The term inside the square root, $b^2 - 4ac$, is known as the discriminant. The discriminant provides valuable information about the nature of the solutions. If the discriminant is positive, there are two distinct real solutions. If it is zero, there is exactly one real solution (a repeated root). If it is negative, there are two complex solutions. The quadratic formula allows us to solve equations that might be difficult or impossible to factorize using traditional methods. It is a universal tool applicable to any quadratic equation, regardless of the nature of its coefficients. The $\pm$ symbol in the formula indicates that there are typically two solutions, one obtained by adding the square root term and the other by subtracting it. These solutions represent the points where the parabola described by the quadratic equation intersects the x-axis. The ability to determine these intersections is crucial in many mathematical and scientific contexts. Furthermore, understanding the quadratic formula enhances one's ability to manipulate and solve algebraic equations, providing a deeper insight into mathematical problem-solving. It bridges the gap between abstract algebraic concepts and practical applications, making it an indispensable tool in both academic and professional settings.

Identifying Coefficients in $x^2 - 12x + 61 = 0$

Before applying the quadratic formula, it is essential to correctly identify the coefficients $a$, $b$, and $c$ in the given quadratic equation $x^2 - 12x + 61 = 0$. Comparing this equation to the general form $ax^2 + bx + c = 0$, we can easily determine the values. The coefficient $a$ is the number in front of the $x^2$ term. In this case, $a = 1$, since $x^2$ is the same as $1x^2$. The coefficient $b$ is the number in front of the $x$ term. Here, $b = -12$. It is crucial to include the negative sign, as it affects the outcome of the formula. The coefficient $c$ is the constant term, which is the number without any $x$ attached. In this equation, $c = 61$. Accurate identification of these coefficients is the foundation for correctly applying the quadratic formula. A mistake in this step can lead to incorrect solutions, highlighting the importance of careful attention to detail. Once the coefficients are identified, they can be substituted into the quadratic formula to calculate the roots of the equation. This process turns an abstract formula into a practical tool for solving a specific problem. Moreover, understanding how to identify coefficients is a fundamental skill in algebra, applicable to a wide range of mathematical problems beyond just quadratic equations. It reinforces the connection between the symbolic representation of equations and their numerical components, fostering a deeper comprehension of algebraic concepts.

Applying the Quadratic Formula

Now that we have identified the coefficients $a = 1$, $b = -12$, and $c = 61$, we can substitute these values into the quadratic formula:

$x = \frac{-b pm psqrt{b^2 - 4ac}}{2a}$

Substituting the values, we get:

$x = \frac{-(-12) pm psqrt{(-12)^2 - 4(1)(61)}}{2(1)}$

First, simplify the expression step by step. The negation of -12 is 12, so we have:

$x = \frac{12 pm psqrt{(-12)^2 - 4(1)(61)}}{2}$

Next, calculate $(-12)^2$, which equals 144. Then, compute $4(1)(61)$, which equals 244. So the equation becomes:

$x = \frac{12 pm psqrt{144 - 244}}{2}$

Subtract 244 from 144 to get -100:

$x = \frac{12 pm psqrt{-100}}{2}$

Now we encounter the square root of a negative number, which indicates that the solutions will be complex. Recall that the square root of -1 is represented by the imaginary unit $i$, so $\sqrt{-100} = \sqrt{100} imes \sqrt{-1} = 10i$. Substituting this, we have:

$x = \frac{12 pm 10i}{2}$

Finally, divide both terms in the numerator by 2:

$x = 6 pm 5i$

Thus, the solutions are $x = 6 + 5i$ and $x = 6 - 5i$. This step-by-step application of the quadratic formula demonstrates the process of finding the solutions, even when they are complex. It reinforces the importance of careful calculation and attention to signs and operations. Furthermore, it highlights the power of the quadratic formula in solving a wide range of quadratic equations, regardless of the nature of their roots. Each step in the process builds upon the previous one, ultimately leading to the solutions. This detailed approach ensures a thorough understanding of how the quadratic formula works and how to apply it effectively.

Understanding Complex Solutions

In the previous section, we found the solutions to the equation $x^2 - 12x + 61 = 0$ to be $x = 6 + 5i$ and $x = 6 - 5i$. These solutions are complex numbers, which consist of a real part and an imaginary part. A complex number is generally expressed in the form $a + bi$, where $a$ is the real part and $bi$ is the imaginary part, with $i$ being the imaginary unit ($\sqrt{-1}$). In our solutions, for $x = 6 + 5i$, the real part is 6 and the imaginary part is $5i$. Similarly, for $x = 6 - 5i$, the real part is 6 and the imaginary part is $-5i$. The presence of complex solutions indicates that the parabola represented by the quadratic equation does not intersect the x-axis. Graphically, this means the parabola either opens upwards and lies entirely above the x-axis or opens downwards and lies entirely below the x-axis. The fact that the discriminant ($b^2 - 4ac$) is negative confirms this, as the square root of a negative number results in an imaginary component. Understanding complex solutions is crucial in many areas of mathematics and physics, including electrical engineering, quantum mechanics, and fluid dynamics. These solutions, though not real, are perfectly valid within the complex number system and provide valuable insights into the behavior of the quadratic equation. Furthermore, the complex solutions always occur in conjugate pairs (i.e., $a + bi$ and $a - bi$), which is a characteristic property of quadratic equations with real coefficients. This understanding deepens one's knowledge of the nature of quadratic equation solutions and their implications in various mathematical and scientific contexts.

Conclusion

In this comprehensive guide, we have demonstrated how to solve the quadratic equation $x^2 - 12x + 61 = 0$ using the quadratic formula. We began by introducing the formula and its significance, then moved on to identifying the coefficients $a$, $b$, and $c$ in the given equation. We meticulously applied the formula, simplifying each step to arrive at the complex solutions $x = 6 + 5i$ and $x = 6 - 5i$. We also discussed the nature of these complex solutions, highlighting their real and imaginary parts and their implications for the graph of the quadratic equation. Understanding the quadratic formula is a fundamental skill in algebra, and this guide aimed to provide a clear and thorough explanation of its application. By mastering this method, you can confidently solve a wide range of quadratic equations, including those with real and complex solutions. The ability to solve quadratic equations is not only essential for academic success but also valuable in various practical applications across different fields. This guide has equipped you with the knowledge and steps necessary to tackle quadratic equations effectively. The quadratic formula serves as a powerful tool, ensuring solutions even when factorization is not straightforward. This concludes our exploration of solving $x^2 - 12x + 61 = 0$ using the quadratic formula, solidifying your understanding of this critical algebraic technique. Remember, practice is key to mastery, so continue to apply this knowledge to various problems to enhance your skills further.